Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 27 October 2018 (MN LATEX style file v2.2)

Activation of the Blandford-Znajek mechanism incollapsing stars

Serguei S. Komissarov,1 Maxim V. Barkov,1,2?1Department of Applied Mathematics, The University of Leeds, Leeds, LS2 9GT2Space Research Institute, 84/32 Profsoyuznaya Street, Moscow 117997, Russia

Received/Accepted

ABSTRACTCollapse of massive stars may result in formation of accreting black holes in their inte-rior. The accreting stellar matter may advect substantial magnetic flux onto the blackhole and promote release of its rotational energy via magnetic stresses (the Blandford-Znajek mechanism). In this paper we explore whether this process can explain thestellar explosions and relativistic jets associated with long Gamma-ray-bursts. In par-ticularly, we show that the Blandford-Znajek mechanism is activated when the restmass-energy density of matter drops below the energy density of magnetic field in thevery vicinity of the black hole (within its ergosphere). We also discuss whether sucha strong magnetic field is in conflict with the rapid rotation of stellar core required inthe collapsar model and suggest that the conflict can be avoided if the progenitor staris a component of close binary. In this case the stellar rotation can be sustained viaspin-orbital interaction. In an alternative scenario the magnetic field is generated inthe accretion disk but in this case the magnetic flux through the black hole ergosphereis not expected to be sufficiently high to explain the energetics of hypernovae by theBZ mechanism alone. However, this energy deficit can be recovered via additionalpower provided by the disk.

Key words: black hole physics – supernovae: general – gamma-rays: bursts – meth-ods: numerical – MHD – relativity

1 INTRODUCTION

The most popular model for the central engines of longGamma-ray-burst (GRB) jets is based on the “failed su-pernova” scenario of stellar collapse, or “collapsar”, wherethe iron core of progenitor star forms a BH (Woosley 1993).If the specific angular momentum in the equatorial part ofprogenitor exceeds that of the marginally bound orbit of theBH then the collapse becomes highly anisotropic. While inthe polar region it may proceed more or less uninhibited, atleast for a while, the equatorial layers form dense and mas-sive accretion disk. The gravitational energy released in thisdisk can be very large, more then sufficient to overturn thecollapse of outer layers and drive GRB outflows (MacFadyen& Woosley 1999). A similar configuration can be producedvia inspiraling of a BH or a neutron star into the compan-ion star during the common envelope phase of a close binary(Tutukov & Iungelson 1979; Frayer & Woosley 1998; Zhang& Fryer 2001). In any case, the observed duration of longGRBs, between 2 and 1000 seconds, imposes a strong con-straint on the size of progenitors - their light crossing time

? email: serguei@maths.leeds.ac.uk, bmv@maths.leeds.ac.uk

should be significantly shorter then the burst duration. Thisimplies that the progenitors must be compact stars strippedof their hydrogen envelopes. This conclusion agrees with theabsence of hydrogen lines in the spectra of supernovae iden-tified with GRBs (Moreover, since some of these supernovaeare of Type Ic their progenitors may have lost their heliumenvelopes as well.). Massive solitary stars can lose their en-velopes via strong stellar winds. However, this may lead tounacceptably low rotation rates by the time of stellar col-lapse (Heger et al. 2005). In the case of binary system theloss of envelope can be caused by the gravitational interac-tion with companion.

The currently most popular mechanism of poweringGRB jets is the heating via annihilation of neutrinos andanti-neutrinos produced in the disk (MacFadyen & Woosley1999). The energy deposited in this way is a strong func-tion of the mass accretion rate which must be above '0.05Ms−1 in order to agree with the observational con-straints on the energetics of GRBs (Popham et al. 1999).Such high accretion rates can be provided in the collapsarscenario and the numerical simulations by MacFadyen &Woosley (1999) and Aloy et al. (2000) have demonstratedthat sufficiently large energy deposition in the polar region

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above the disk may indeed result in fast collimated jets.However, we have to wait for simulations with proper im-plementation of neutrino transport before making final con-clusion on the suitability of this model – the long and com-plicated history of numerical studies of neutrino-driven su-pernova explosions teaches us to be cautious. The first at-tempt to include neutrino transport in collapsar simulationswas made by Nagataki et al. (2007) and they did not seeneutrino-driven polar jets - the heating due to neutrino an-nihilation could not overcome the cooling due to neutrinoemission.1 In any case, it is unlikely that this model can ex-plain the bursts that are longer than 100s as by this timethe mass accretion rate is expected to drop significantly.For example the accretion rate in the simulations by Mac-Fadyen & Woosley (1999) has already drifted below the crit-ical M ' 0.05Ms−1 by t = 20s.

The two alternatives to the neutrino mechanism whichare also frequently mentioned in connection with GRBs arethe magnetic braking of the accretion disk (Blandford &Payne 1982; Narayan et al. 1992; Meszaros & Rees 1997;Proga et al. 2003; Uzdensky & MacFadyen 2006) and themagnetic braking of the central black hole (Blandford &Znajek 1977; Meszaros & Rees 1997; Lee et al. 2000; McK-inney 2006; Barkov & Komissarov 2008a,b).

A number of groups have studied the potential of mag-netic mechanism in the collapsar scenario using NewtonianMHD codes and implementing the Paczynski-Witta poten-tial in order to approximate the gravitational field of cen-tral BH (Proga et al. 2003; Fujimoto et al. 2006; Nagatakiet al. 2007). In this approach it is impossible to capturethe Blandford-Znajek effect and only the magnetic brakingof accretion disk can be investigated. The general conclu-sion of these studies is that the accretion disk can launchmagnetically-driven jets provided the magnetic field in theprogenitor core is sufficiently strong. This field is further am-plified in the disk, partly due to simple winding of poloidalcomponent and partly due to the magneto-rotational insta-bility (MRI), until the magnetic pressure becomes very largeand pushes out the surface layers of the disk. In particular,Proga et al. (2003) studied the collapse of a star with asimilar structure to that considered MacFadyen & Woosley(1999) and with purely monopole initial magnetic field ofstrength B = 2 × 1014G at r = 3rg. The corresponding to-tal magnetic flux, Ψh ' 1026G cm2, is in fact within theobservational constraints on the field of magnetic stars, in-cluding white dwarfs (Schmidt et al. 2003), Ap-stars (Fer-rario & Wickramasinghe 2005) and massive O-stars Donatiet al. (2002). They used realistic equation of state and in-cluded the neutrino cooling but not the heating. The simu-lations lasted up to t ' 0.28s in physical time during whicha Poynting-dominated polar jet has developed in the solu-tion. The jet power seemed to show strong systematic de-cline, from ' 1051erg s−1 down to ' 1050erg s−1 or even less,which might signal transient phenomenon.

Sekiguchi & Shibata (2007) studied the collapse of ro-

1 However, they have simulated only the very initial stages of

the collapsar evolution, t < 2.2s, whereas the jet simulation ofMacFadyen & Woosley (1999) started at t = 7s when the density

in the polar regions above the black hole has dropped down to

106g cm−3 leading to much lower cooling rates.

tating stellar cores and formation of BHs and their disks inthe collapsar scenario using the full GRMHD approxima-tion. Their simulations show powerful explosions soon afterthe accretion disk is formed and the free falling plasma ofcollapsing star begins to collide with this disk. However,they did not account for the neutrino cooling and the en-ergy losses due to photo-dissociation of atomic nuclei sothese explosions could be similar in nature to the “suc-cessful” prompt explosions of early supernova simulations(Bethe 1990). Mizuno et al. (2004a,b) carried out GRMHDsimulations in the time-independent space-time of a cen-tral BH. Their computational domain did not capture theBH ergosphere and thus they could not study the role ofthe Blandford-Znajek effect (Komissarov 2004a, 2008). Inaddition, the energy gains/losses due to the neutrino heat-ing/cooling were not included and the equation of state(EOS) was a simple polytrope. The simulations run for arather short time, ' 280rg/c where rg = GM/c2, and jetswere formed almost immediately, presumably due to the ex-tremely strong initial magnetic field.

The original theory of the Blandford-Znajek mechanismwas developed within the framework of magnetodynamics(the singular limit of RMHD approximation where the in-ertia of plasma particles is fully ignored). The pioneeringnumerical studies of this mechanism have shown that it canalso operate within the full RMHD approximation, at leastin the regime where the energy density of matter is muchsmaller compared to that of the electromagnetic field, andthat it can be captured by modern numerical techniques(Komissarov 2004b; Koide 2004; McKinney & Gammie 2004;Nagataki 2009). However, no systematic numerical study hasbeen carried out in order to establish the exact conditionsfor activation of the BZ mechanism.

The first simulations demonstrating the potential of theBZ mechanism in the collapsar scenario were carried out byMcKinney (2006) who also used simple polytropic EOS andconsidered purely adiabatic case, but captured the effects ofblack hole ergosphere. The initial configuration included atorus with poloidal magnetic and a free-falling stellar enve-lope. The results show ultrarelativistic jets with the Lorentzfactor up to 10 emerging from the black hole indicating thatthe Blandford-Znajek effect may play a key role in the pro-duction of GRB jets. Barkov & Komissarov (2008a,b) addeda realistic EOS and included the energy losses due to neu-trino emission (assuming optically thin regime) and photo-dissociation of nuclei. They also reported the developmentof relativistic outflows powered by the central black hole viathe Blandford-Znajek mechanism (the magnetic braking ofaccretion disk did not seem to contribute much to the jetpower though).

The black hole-driven magnetic stellar explosions andrelativistic jets observed in these exploratory computer sim-ulations invite a deeper analysis of this model. Obviously,such an outcome cannot be a generic feature of core-collapsesupernovae. Indeed, only a very small fraction of SNe Ib/Icseem to produce GRBs (Piran 2005; Woosley & Bloom2006). So what are the relevant conditions and can theybe reproduced as the results of stellar evolution? These arethe main issues we address in this paper.

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Activation of the BZ mechanism 3

2 BLANDFORD-ZNAJEK MECHANISM ANDGRBS

The rotational energy of Kerr black hole is

Erot = Mhc2f1(a) ' 1.8×1054f1(a)

(Mh

M

)erg, (1)

where

f1(a) = 1− 1

2

[(1 +

√1− a2

)2

+ a2

]1/2

,

Mh is the BH mass and a ∈ [0, 1) is its dimensionless ro-tation parameter. For Mbh = 2M and a = 0.9 this givesthe enormous value of Erot ' 5 × 1053erg which is aboutfifty times higher than the rotational energy of a millisec-ond pulsar and well above of what is needed to explain theenergy of GRBs and associated hypernovae. Even for a rel-atively slowly rotating black hole with a = 0.1 this is stilla respectable number, 2.3×1051erg. Moreover, because inall versions of the collapsar model the black hole spin isaligned with the spin of accretion disk this energy reserveris continuously replenished via accretion. Thus, as far as theavailability of energy is concerned the black hole model looksvery promising indeed.

The energy release rate is usually estimated using theBlandford-Znajek power for the case of force-free monopolemagnetosphere

EBZ =1

6c

(ΩhΨh

8π

)2

, (2)

where Ωh is the angular velocity of the BH and Ψh is themagnetic flux threading one hemisphere of the BH horizon(Here it is assumed that the angular velocity of the BH mag-netosphere Ω = 0.5Ωh.). This formula is quite accurate notonly for slowly rotating black holes considered in Blandford& Znajek (1977) but also for rapidly rotating BHs (Komis-sarov 2001). In application to the collapsar problem it givesus the following estimate

EBZ = 1.4× 1051f2(a)Ψ2h,27

(Mh

M

)−2

erg s−1, (3)

where

f2(a) = a2(

1 +√

1− a2

)−2

,

and Ψh,27 = Ψh/1027G cm2. One can see that the powerof BZ-mechanism is rather sensitive to black hole’s massand magnetic flux. Since, the black hole mass is likely tobe ≥ 3M the observed energetics of hypernovae and longGRBs requires Ψh,27 ' 1. This value is comparable withthe maximal surface flux observed in magnetic stars, Apstars, magnetic white dwarfs, and magnetars (e.g. Ferrario& Wickramasinghe 2005). Thus, the magnetic field of GRBcentral engine may well be the original field of the progenitorstar.

The estimates (1,3) show that braking of black holesalone can explain the energetics of GRBs and this is why thismechanism is often mentioned in the literature on GRBs.However, there is another issue to take into consideration.The equation (2) is obtained in the limit where the inertiaof magnetospheric plasma and, to large degree, its gravita-tional attraction towards the black hole are ignored. In con-trast, the mass density of plasma in the collapsing star may

be rather high and has to be taken into account. For exam-ple, the magnetohydrodynamic waves may become trappedin the accretion flow and unable to propagate outwards. Insuch a case, it would not be possible to extract magneticallyany of the black hole rotational energy and drive stellar ex-plosion irrespectively of how high this energy is. Since theBZ mechanism can only operate within the black hole ergo-sphere (e.g. Komissarov 2004a, 2008), the magnetohydrody-namic waves must at least be able to cross the ergosphere inthe outward direction for its activation. This agrees with theconclusion reached in Takahashi et al. (1990) that the Alfvensurface of a steady-state ingoing wind must be located in-side the ergosphere to have outgoing direction of the totalenergy flux. Thus we should consider the following conditionfor activation of the BZ-mechanism in accretion flows: theAlfven speed has to exceed the local free fall speed at theergosphere,

va > vf .

In order to simplify calculations let us apply the Newtonianexpressions for both speeds. The Alfven speed is

v2a =

B2

4πρ,

where B is the magnetic field strength and ρ is the plasmamass density, whereas the free-fall speed is

v2f =

2GMh

r.

At r = 2rg = 2GM/c2 the local criticality condition reads

βρ =4πρc2

B2< 1. (4)

That is the energy density of magnetic field has to exceedthat of plasma in the vicinity the black hole.

For spherically symmetric flows the condition va > vfcan be written as a constraint on the mass accretion rate,M , and the flux of radial magnetic field, Ψ,

Ψ

2πr√Mvf (r)

> 1.

Since vf ∝ r−1/2 this condition is bound not to be satisfiedat large r but we only need to apply it at the ergosphere.Using r ' 2rg we then obtain

κ > κc, (5)

where

κ =Ψh

4πrg√Mc

, (6)

and κc = 1. In fact, the critical value of κ must dependon the black hole spin and tend to infinity for small a. TheNewtonian analysis cannot capture this effect. The relativis-tic one, on the other hand, appears rather complicated (Ca-menzind 1989; Takahashi et al. 1990). Fortunately, nowa-days this issue can be investigated via numerical simula-tions. Another, interesting and important issue for time-dependent simulations is whether there is only one bifurca-tion separating solutions with switched-on and switched-offBZ mechanism or the transition is reacher and allows morecomplicated time-dependent solutions, e.g. quasi-periodic orchaotic ones.

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4 S.S. Komissarov & M.V.Barkov

3 TEST SIMULATIONS.

The first type of simulations presented in this paper aredesigned to test the validity of our arguments behind thecriterion for activation of BZ-mechanism which we derivedin Sec.2 using a combination of Newtonian and relativisticphysics. For this purpose we consider a more or less spheri-cal accretion of cold magnetized plasma with vanishing an-gular momentum onto a rotating black hole. In order tofocus on the MHD aspects of the problem we ignore the mi-crophysics important in the collapsar problem and considerplasma with simplified polytropic EOS.

The main details of our numerical method and varioustest simulations are described in Komissarov (1999, 2004b,2006). The only really new feature here is the introductionof HLL-solver which is activated when our linear Riemannsolver fails. This usually occurs in regions of relativisticallyhigh magnetization where the magnetic energy density ex-ceeds that of matter by more than one order of magnitudeor in strong rarefactions. This makes the scheme a little bitmore robust though we are still forced to use a “densityfloor”, that is a lower limit on the value of mass-energy den-sity of matter2.

Both in the numerical scheme and throughout this sec-tion of the paper we utilize geometric units where G = c =Mh = 1. The Kerr spacetime is described using the Kerr-Schild coordinates so the metric form reads

ds2 = gttdt2 + 2gtφdtdφ+ 2gtrdtdr+

gφφdφ2 + 2grφdφdr + grrdr

2 + gθθdθ2,

(7)

where

gtt = ζ − 1, gtφ = −ζa sin2θ, gtr = ζ,

gφφ = Σ sin2 θ/A, grφ = −a sin2 θ(1 + ζ),

grr = 1 + ζ, gθθ = A,

and

A = r2 + a2 cos2θ,

ζ = 2r/A,

Σ = (r2 + a2)2 − a2∆ sin2θ,

∆ = r2 + a2 − 2r.

3.1 Setup

The initial solution has to describe both the flow and themagnetic field. In this case the flow is given by the steady-state solution for unmagnetised cold plasma (dust). At in-finity it is spherically symmetric and has vanishing specificangular momentum and radial velocity. The components of4-velocity of this flow in the coordinate basis of Kerr-Schildcoordinates, ∂ν, are

2 We have found that a total switch to the HLL-solver noticeablydegrades numerical solutions via increasing numerical diffusion

and dissipation. This is particularly noticeable for slowly evolvingflow components, like accretion disks (see also Mignone & Bodo

(2006).

ut = 1 + ζη/(1 + η);

uφ = −a/(A(1 + η));ur = −ζη;

uθ = 0,

(8)

where η =√

(r2 + a2)/2r and its rest mass density is

ρ = ρ+

(r+r

)1

η(9)

where ρ+ is the rest mass density at the outer event horizonradius (see Appendix A).

The initial magnetic field is radial with monopole topol-ogy

Bφ = aBr(A+ 2r)/(A∆ + 2r(r2 + a2));Br = B0 sin θ/

√γ;

Bθ = 0.(10)

The azimuthal component is introduced in order to ensurethat the electromagnetic fluxes of energy and angular mo-mentum vanish (see Appendix B) and, thus, the initial mag-netic configuration is more or less dynamically passive.

The two-dimensional axisymmetric computational do-main is (r0<r<r1)× (0<θ<π), where r0 = 1 + 0.5

√1− a2

and r1 = 500. Notice that the inner boundary is inside ofthe outer event horizon, r+ = 1 +

√1− a2. This can be

done because of the non-singular nature of the horizon inthe Kerr-Schild coordinates. The computational grid is uni-form in θ, where it has 100 cells, and almost uniform in log r,where it has 170 cells. The metric cell sizes are the same inboth directions. The grid is divided into rings such that forany ring the metric size of its outer cells is twice that of itsinner cells. The solution on each ring is advanced with itsown time step, the outer ring time step being twice that ofthe inner ring. At the interface separating cells of differentrings the conservation of mass, energy, and momentum andmagnetic flux is ensured via simple averaging procedure.

At the inner boundary, r = r0, we impose the “non-reflective” free-flow boundary conditions. The use of theseconditions is justified by the fact that this boundary lies in-side the outer event horizon in the region where all physicalwaves move inwards. At the outer boundary, r = r1 we fixthe flow parameters corresponding to the initial solution.In all runs the simulations are terminated well before anystrong perturbation generated near the centre reaches thisboundary. At the polar axis, θ = 0, π, the boundary condi-tions are dictated by the axial symmetry. Namely, quantitiesthat do not vanish on the axis are reflected without changeof sign, whereas quantities that vanish on the axis are re-flected with change of sign.

We consider models which differ only by the rotationrate of black hole, a, and the mass accretion rate, M . Foreach value of a, we first deal with the model with κ ' 1. De-pending on whether the BZ-mechanism is activated in thismodel or not we either increase or reduce ρ+ (and henceM) by the factor of three and compute another model. Weproceed in this fashion until the solution becomes qualita-tively different. Then we compute one more model based onthe mean value of ρ+ for the previous two models. Afterthat we select the closest two models with qualitatively dif-ferent solutions and show them on the bifurcation diagramsummarising the results of our study.

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Activation of the BZ mechanism 5

Figure 1. Free-fall accretion of cold plasma with zero angular momentum onto a Kerr black hole (a = 0.9); the global structure. The

colour images show log10 ρ in dimensionless units and the contours show the magnetic field lines ( To be more specific the contours showthe levels of poloidal magnetic flux function, Ψ. At any point (r, θ) this function gives the magnetic flux through the axisymmetric loop

or radius r passing through this point.). Top panel: model with the activation parameter κ = 1.2 at time t = 2000rg/c. Middle panel:

model with κ = 1.6 and monopole magnetic field at time t = 1000rg/c. Bottom panel: model with κ = 1.6 and split-monopole magneticfield at time t = 1600rg/c. The unit of length is rg .

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6 S.S. Komissarov & M.V.Barkov

Figure 2. Free-fall accretion of cold plasma with zero angular momentum onto a Kerr black hole (a = 0.9); the central regions. The top

panels show the model with κ = 1.2 at t = 2000rg/c; the middle panels show the model with κ = 1.6 and monopole magnetic field att = 1000rg/c, the bottom panels show the model with κ = 1.6 and split-monopole magnetic field at t = 1600rg/c. The left panels show

the rest mass density distribution (ρ for the top panel and log10 ρ for others) and the magnetic field lines. The right panels show the

ratio of the proper rest mass and magnetic energy densities, log10 βρ, and the velocity field (the length of velocity arrow is proportionalto the square root of speed.).

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Activation of the BZ mechanism 7

3.2 Results

The simulations do confirm the anticipated bifurcation andshow that for a wide of parameter a the critical value of κ isindeed close to unity. For example, the model with a = 0.9and κ = 1.2 is qualitatively different from the model themodel with a = 0.9 and κ = 1.6. The first model quicklysettles to a steady state accretion solution (see the top pan-els of fig.1 and fig.2). In fact, on the large-scales, this solutionis almost indistinguishable from the initial solution. On thesmall scales, near the event horizon, the differences becomemore pronounced. In particular, a stationary accretion shockappears just outside of the ergosphere. It is responsible forthe jump in the rest mass density and the kinks of magneticfield lines seen in the top left panel of fig.2. Inside the ergo-sphere the magnetization parameter µ is larger than unitybut only just. The magnetic field lines show no rotation andthus the BZ-mechanism remains completely switched-off.

In contrast, the model with a = 0.9 and κ = 1.6 ex-hibits powerful bipolar outflow which drives a blast waveinto the accreting flow (see the middle panel of fig.1). Thisblast waves overturns the accretion both in the polar and inthe equatorial direction. The black hole develops highly rar-efied magnetosphere (see the middle panels of fig.2) whichrotates with about half of black hole’s angular velocity. Theshocked plasma of accretion flow is prevented from accretiononto the black hole – instead it participates in the large scalecirculations above and below the equatorial plane. The un-physical monopole configuration of magnetic field set for thismodel prevents any escape of magnetic flux from the blackhole. This ensures operation of the BZ-mechanism even ifthe confining accreting envelope is fully dispersed into thesurrounding space.

For a more realistic dipolar configuration one would ex-pect at least partial escape of magnetic flux once the outflowis developed and thus less power in the outflow. However,this is unlikely to effect the value of κc. Indeed, considerthe case with monopole field where the BZ-mechanism re-mains switched off. Then, change the polarity of magneticfield lines in the northern hemi-sphere. Because the speedsof MHD waves and the magnetic stresses remain invariantunder change of magnetic polarity this will have no effecton the flow. For the very same reason, we expect activa-tion of the BZ mechanism for exactly the same value of κ inboth magnetic configurations. In order to verify this conclu-sion we computed models with the initial magnetic field ofsplit-monopole topology and the results fully agree with theexpectations – the critical value of κ remains unaffected butthe power of BZ-mechanism is reduced due partial escapeof magnetic flux from the black hole. The bottom panel offig.1 shows the large scale flow developed in the model witha = 0.9, κ = 1.6, and split-monopole initial magnetic field.The flow pattern is very similar to that of the model withmonopole field but the expansion rate is noticeable lower,approximately by the factor of 1.8 in the polar directionand even more in the equatorial direction. The bottom pan-els of fig.2 show the inner region of this model. One can seethat the accretion is no longer fully halted – the accretingplasma finds its way into the black hole in the equatorialzone and the black hole magnetosphere is confined only tothe polar region of the funnel shape.

The results for a = 0.9 show that not only there is

Figure 3. The bifurcation diagram. The stars show the modelswith the lowest value of κ which still show activation of the BZ

mechanism. The circles show models with the highest value of κ

where the BZ mechanism remains turned off. The pair of pointswith a = 0.9 in the right bottom corner of the diagram correspond

to proper collapsar simulations. The other points show the results

for test simulations.

a bifurcation between the regimes with switched-on andswitched-off BZ process but also that the critical value ofthe control parameter κ, and hence βρ, is indeed close tounity at least for rapidly rotating black holes. The results ofour parametric study, summarised in fig.3, show that in factκc(a) is a relatively weak function of a. It is confined withinthe range (1, 2) for 0.2 < a < 1.0.

4 COLLAPSAR SIMULATIONS

The test simulations described in Sec.3 allow us to deter-mine the basic criteria for activation of the BZ mechanismin principle. However, their set-up is rather artificial as inmany astrophysical applications the accreting plasma hassufficient angular momentum to ensure formation of accre-tion disk and thus to render the approximation of sphericalaccretion unsuitable. This anisotropy of mass inflow is likelyto make the integral activation condition (5) less helpful. Onthe other hand, one would expect the local condition con-dition (4) to be more or less robust. In order to check thiswe analysed the results obtained in our current numericalstudy of magnetic collapsar model.

4.1 Computational grid and boundary conditions

Like in the test simulations we use the Kerr metric inKerr-Schild coordinates, φ, r, θ. We selected Mh = 3M

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8 S.S. Komissarov & M.V.Barkov

and the rather optimistic value of a = 0.9 for the cen-tral black hole. The two-dimensional axisymmetric compu-tational domain is (r0 < r < r1) × (0 < θ < π), with r0 =(1 + 0.5

√1− a2) rg = 5.4 km and r1 = 5700 rg = 25000 km.

The computational grid is of the same type as in the testsimulations (Sec.3.1) but with more cells, 450 × 180. Theboundary conditions are also the same as in the test simu-lations.

4.2 Microphysics

For these simulations we use realistic equation of state(EOS) that takes into account the contributions from radi-ation, lepton gas including pair plasma, and non-degeneratenuclei (hydrogen, helium, and oxygen). This is achievedvia incorporation of the EOS code HELM (Timmes &Swesty 2000), which can be downloaded from the web-sitehttp://www.cococubed.com/code pages/eos.shtml.

The neutrino cooling is computed assuming opti-cally thin regime and takes into account URCA-processes(Ivanova et al. 1969), pair annihilation, photo-production,and plasma emission (Schinder et al. 1987), as well as syn-chrotron neutrino emission (Bezchastnov et al. 1997). Infact, URCA-processes strongly dominate over other mech-anisms in this problem. Photo-disintegration of nuclei isincluded via modification of EOS following the prescrip-tion given in Ardeljan et al. (2005). The equation for massfraction of free nucleons is adopted from Woosley & Baron(1992). We have not included the radiative heating due toannihilation of neutrinos and antineutrinos produced in theaccretion disk mainly because this requires elaborate andtime consuming calculations of neutrino transport.

4.3 Model of collapsing star

The collapsing star is introduced using the simple free-fallmodel by Bethe (1990). In this model it is assumed that im-mediately before the collapse the mass distribution in thestar satisfies the law ρr3 =const and that the infall pro-ceeds with the free-fall speed in the gravitational field of thecentral black hole. This gives us the following radial velocity

vrff = (2GM/r)1/2 (11)

and mass density

ρ = C1 × 107(ts1 s

)−1 ( r

107cm

)−3/2

g cm−3 (12)

for the collapsing star, where C1 is a constant that dependson the star mass (Bethe 1990) and ts is the time since thestart of collapse. The corresponding accretion rate and rampressure are

M = 0.056 C1

(M

3M

)1/2 (ts1 s

)−1

Ms−1, (13)

pram = 7.9×1026C1M

3M

(ts1 s

)−1 ( r

107cm

)−5/2 g

cm s2(14)

respectively. For a core of radius rc = 109cm and mass3.0M the collapse duration can be estimated as tc =2rc/3vff ' 0.6s. Since, in this study we explore the pos-sibility of early explosion, that is soon after the core col-

lapse, we set ts = 1s. Because GRBs are currently associ-ated with more massive progenitors we consider C1 = 3, 9.This gives us the accretion rates of M ' 0.166Ms−1 and

M ' 0.5Ms−1 respectively.On top of this we endow the free-falling plasma with

angular momentum and poloidal magnetic field. The angularmomentum distribution is

l = sin θ

l0(r sin θ/rl)

2 if r sin θ < rll0 if r sin θ > rl

, (15)

where rl = 6300km and l0 = 1017cm2s−1.3

4.4 Initial magnetic field

The origin of magnetic field is probably the most impor-tant and difficult issue of the magnetic model of GRBs. Onepossibility is that this field already exists in the progeni-tor and during the collapse it is simply advected onto theblack hole. The recent numerical studies indicate that thepoloidal and the azimuthal components of the progenitorfield should have similar strengths (Braithwaite & Spruit2004). The magnetic flux freezing argument then suggeststhat during the collapse the poloidal field becomes dominantand thus in our simulations we may ignore the initial az-imuthal field. As to the initial poloidal component we adoptthe solution for a uniformly magnetized sphere in vacuum4.To be more specific, we assume that inside the sphere ofradius rm = 4500 km the magnetic field is uniform and hasstrength of either B0 = 3× 109, 1010, or 3× 1010G, whereasoutside it is described by the solution for magnetic dipole.The selected values for the field strength allow to capturethe BZ bifurcation. Given the limitations of axisymmetrywe can only consider the case of aligned magnetic and rota-tional axes.

4.5 Results

Table 1 describes the key parameters of six models inves-tigated in this study and also summarises the outcome ofsimulations. One of the parameters, Ψmax, is the total mag-netic flux enclosed by the equator of the uniformly magne-tized sphere in the initial solution. Obviously, this gives usthe highest magnetic flux that can be accumulated by theblack hole in the simulations. Whether this maximum valueis actually reached at some point depends on the progen-itor rotation. For slow rotation the whole of the uniformmagnetized core can be swallowed by the hole before thedevelopment of accretion disk. At this point Ψh equals toΨmax and then it begins to decline as the result of advec-tion of the oppositely directed parts of dipolar loops (Theseloops are clearly seen in fig.5). For fast rotation the accretiondisk forms before Ψh reaches Ψmax and after this point the

3 The intention was to set up a solid body rotation within the

cylindrical radius rl, however, by mistake an additional factor,sin θ, was introduced in the start-up models (The same mistake

was made in the simulations described in Barkov & Komissarov

(2008a).) Although somewhat embarrassing, this does not affectthe main results of this study.4 Thus, we ignore the magnetic field that is already threading

the black hole by the start of simulations.

c© 0000 RAS, MNRAS 000, 000–000

Activation of the BZ mechanism 9

Figure 4. Model B close to the explosion, at t = 237ms. Top left: log10 ρ and the poloidal velocity; Top right: the angular velocity of

plasma and the poloidal velocity; Bottom left: the ratio of azimuthal and poloidal components of magnetic field, log10(|Bφ|/|Bp|), andthe magnetic field lines; Bottom right: log10 βρ, the magnetic field lines, and the poloidal velocity field. The length of velocity arrows is

proportional to the square root of speed.

c© 0000 RAS, MNRAS 000, 000–000

10 S.S. Komissarov & M.V.Barkov

name B0,10 C1 Ψmax27 κmax Exp Ψh,27 E51

A 3 3 19.1 1.07 Yes 11.6 9.6

B 3 9 19.1 0.62 Yes 13.1 12.7

C 1 3 6.67 0.36 Yes 2.80 0.4D 1 9 6.67 0.21 No

E 0.3 3 1.91 0.11 No

F 0.3 9 1.91 0.06 No

Table 1. Summary of numerical models in collapsar simulations.B0,10 is the initial magnetic field strength in units of 1010G, C1

is the density parameter in eq.12, Ψmax27 is the maximum value of

magnetic flux that can be accumulated by the black hole in unitsof 1027G cm2, κmax is the value of κ corresponding to Ψmax27 ,

“Exp” is the explosion indicator, Ψh,27 is the actual magnetic

flux accumulated by the black hole by the time of explosion, E51

is the mean power of the BZ mechanism in units of 1051erg/s.

growth of Ψh slows down noticeably as the disk accretion ismuch slower compared to the free-fall accretion. Once theoppositely directed parts of magnetic loops begin to crossthe accretion shock and enter the turbulent domain that sur-rounds the disk the annihilation of poloidal magnetic field inthis domain becomes an important factor. As the result themaximum value of magnetic flux available in the solutionbegins to decrease before Ψh reaches Ψmax.

κmax is the value of parameter κ computed using Ψmax

instead of the actual value of black hole flux in eq.5. Quickinspection shows that the critical value of κ based on thevalues of κmax is about κc = 0.3 (see models C and D intable 1.), almost five time lower than that in the test sim-ulations of spherical collapse. In fact, the actual value ofmagnetic flux accumulated by the black hole, Πh, by thetime of explosion is always noticeably lower than κmax (seetable 1). In model C it is 2.4 times lower thus driving theactual critical value of κ down to the value around 0.1. Thisresult is undoubtedly connected to the highly anisotropicnature of the flow in the close vicinity of the black holethat develops in the simulations. Most importantly, there isa clear separation of the mass and magnetic fluxes: while alarge fraction of mass is accreted through the accretion disk,the bulk of magnetic field lines connected to the black holeoccupy the space above (and below) the disc (see fig.4).

All models considered in this study first pass throughthe passive phase of almost spherical accretion. During thisphase the black hole accumulates magnetic flux but asΨmax is not sufficiently high the Blandford-Znajek mech-anism remains switched-off. Then, as the angular momen-tum of plasma approaching the black hole exceeds that ofthe marginally bound orbit, the accretion disk forms in theequatorial plane and an accretion shock separates from itssurface. This shock is best seen in the density plot of fig.4where it appears as a sharp circular boundary between thelight blue and dark blue areas of radius ' 30rg ' 135km. Infact, the accretion shock oscillates in a manner remindingthe oscillations found earlier in supernova simulations (e.g.Blondin et al. 2003; Buras et al. 2006; Scheck et al. 2008),as well as in simulations of accretion flows onto black holes(Nagakura & Yamada 2009), and attributed to the so-calledStationary Accretion Shock Instability (SASI). Most likely,the physical origin of shock oscillations in our simulations isthe same but we have not explored this issue in details.

During this second phase the differential rotation re-sults in winding up of the magnetic field in the disk and itscorona where the azimuthal component of magnetic field be-gins to dominate (top-right and bottom-left panels of fig.4).The corona acts as a “magnetic shield” deflecting the plasmacoming at intermediate polar angles towards either the equa-torial plane or the polar axis (see fig.4). In models D,E andF the second phase continues till the end of run and theBlandford-Znajek process remains switch-off permanently.There are no clear indications suggesting that this phasemay end soon although the accretion disk grows monotoni-cally in radius and this may eventually bring about a bifur-cation.

In models A,B,C after several oscillations the solutionenters the third phase which is characterized by a non-stopexpansion of the accretion shock which then turns into ablast wave. During this transition the black hole developsa low mass density magnetosphere which occupies the ax-ial funnel through which the rotational energy of the blackhole is transported away away in the form of Poynting fluxand powers the blast wave. The typical large-scale structureof the the solution during the third phase is shown in fig.5(see also Barkov & Komissarov (2008a) ). It is rather simi-lar to the structure observed in the simulations of sphericalaccretion in the case of split-monopole magnetic field (seeSec.3.2).

Figure 4 shows the solution for model B at time t =237ms, which is close to the end of the second phase, andhelps to understand how exactly the BZ mechanism is ac-tivated in our simulations. At this stage, the accretion diskis well developed and accounts for ' 50− 60% of mass fluxthrough the event horizon whereas most of the magneticfield lines threading the horizon avoid both the disk andcorona that surrounds the disk and contains predominantlyazimuthal magnetic field. The strong turbulent motion inthe disk and its corona must be the reason for this expul-sion of magnetic flux (Zel’dovich 1957; Radler 1968; Tao etal. 1998). In any case, the observed separation of magneticand mass fluxes results in a rather inhomogeneous magne-tization of plasma near the black hole. Most importantly,there are regions that have much higher local magnetizationcompared to the case of spherical accretion with the samevalue of integral control parameter κ.

Moreover, the magnetic shield action of the disk coronacan promote further local increase of magnetization. Indeed,the corona can deflect plasma passing through the accretionshock at intermediate latitudes in the direction which doesnot necessarily coincide with the local direction of the mag-netic field. For example, in figure 4 the stagnation pointof post-shock flow in the upper hemisphere is located at(r, z) ' (25, 20)rg whereas the separation between the mag-netic field lines connected to the hole and to the disk occursat much lower latitude, around (r, z) = (28, 10)rg. Thus,the direct mass loading of the field lines entering the shockbetween these points with is currently suspended. On theother hand, at the other end of these field lines, where theypenetrate the event horizon, their unloading continues asusual. As the result, the magnetization along these linesincreases allowing to reach the local criticality condition,βρ < 1 (see the intense blue regions in the left panel of fig.4).Once this condition is satisfied the BZ mechanism turns on,at this point only very locally and the initial rotation rate

c© 0000 RAS, MNRAS 000, 000–000

Activation of the BZ mechanism 11

Figure 5. Model B soon after the explosion. The figure shows

the solution at t = 332ms. The colour image gives log10 ρ and thecontours show the magnetic field lines.

of the field lines is significantly lower then 0.5Ωh. However,an additional magnetic energy is now being pumped intothe shield via the BZ mechanism promoting its expansionand increasing its efficiency as a flow deflector. This causesfurther increase of the magnetization at its base, further in-crease of magnetosphere’s rotation rate and thus higher BZpower and so on. The process develops in a runaway fashion.

Figure 6 shows the magnetization parameter βρ at r =2rg as a function of the polar angle for model B (at the sametime as in fig.4) and for model D at a similar time. One cansee that for model D the magnetization is lower. In fact, thelocal criticality condition, βρ < 1, is not satisfied anywhere

Figure 6. The magnetization parameter βρ = 4πρc2/B2 at r =2rg for models B at t = 237ms (solid line) and D at t = 360ms

(dashed line).

for this model and this explains why the explosion is notproduced in this model5.

5 DISCUSSION

5.1 The strength and origin of magnetic field.

There seems to be only two possible origins for the magneticfield in the collapsar problem. Either this is a fossil field ofthe progenitor star of the field generated in the accretiondisk and its corona. Here we consider both possibilities.

The blunt application of the integral condition (5) foractivation of the Blandford-Znajek mechanism gives us thecritical value of magnetic flux accumulated by the black hole

Ψh,c = 4.3× 1028

(Mh

3M

)(Ms

M

)1/2

G cm2.

Although the strength of magnetic field in the interior ofmassive stars which end up as failed supernovae is not knownthis is uncomfortably high compared to fluxes deduced fromthe observations of “magnetic stars” and rises doubts aboutthe fossil hypothesis. However, the actual values of the crit-ical magnetic flux found in our simulations are significantlylower. As we have already commented in Sec.4 this is dueto the anisotropy of accretion in the collapsar model. Thisanisotropy is expected to increase with time as more distantparts of the progenitor with higher angular momentum enter

5 These results also suggest that both numerical diffusion andresistivity reduce chances of successful explosion in computer sim-ulations via smoothing the distributions of mass and magnetic

field and thus decreasing inhomogeneity of βρ.

c© 0000 RAS, MNRAS 000, 000–000

12 S.S. Komissarov & M.V.Barkov

the accretion shock (we could not study this late phase of thecollapse, t 1 s, because of various technical issues.). Dueto the higher angular momentum the predominant motiondownstream of the shock will be towards the equator andthis should result in (i) further reduction of the mass fluxand the ram pressure of accreting plasma in the polar regionand (ii) more effective mass-unloading of magnetic field linesentering the black hole (see Sec.4.5). Both effects promotehigher plasma magnetization in the polar region around theblack hole which may eventually exceed the local criticalitycondition (4) even in models with Ψh < 1027Gcm2, makingthe fossil hypothesis more plausible.

Our results seem at odds with the Newtonian simula-tions by Proga et al. (2003) who obtain magnetically drivenexplosions powered by the disk wind already within 300 msin physical time since the start of simulations for a modelwith similar mass accretion rate, M ' 0.29M/s, but muchlower magnetic field, Ψ ' 1026G cm2. Since their initialmodel is similar to that of MacFadyen & Woosley (1999) thedifference cannot be attributed to the development of highlyrarefied polar channel seen in the simulations of MacFadyen& Woosley (1999) as this occurs only several seconds later.Other possible reasons include differences in the framework,details of the initial solutions, resolution, implementation ofmicrophysics6 etc. Hopefully, simulations by other groupswill soon clarify this issue.

Another factor that is likely to reduce the constrainton the strength of the fossil field, and the one that is com-pletely ignored in our simulations, is the plasma heating dueto neutrino-antineutrino annihilation. In the most extremescenario this heating completely overpowers accretion in thepolar direction and drives a relativistic GRB jet (MacFadyen& Woosley 1999). Although this is an attractive model ofGRB explosions by itself we also note that the creation oflow density funnel in this model helps to activate the BZmechanism along the magnetic field lines that are advectedinside this funnel. Thus, the neutrino heating and BZ mech-anism may work hand in hand in driving GRBs and hyper-novae.

Even in failed supernovae the stellar core collapse doesnot lead directly to formation of a black hole but still pro-ceeds through the proto-neutron star (PNS) phase first. Ifthe period of PNS is very short, around several millisecons,its rotational energy is sufficient to strongly modify the di-rection of collapse and the initial conditions for the blackhole phase. In particular, the magnetic stresses due to ei-ther fossil or MRI-generated magnetic field can drive pow-erful bipolar outflows (Moiseenko et al. 2006; Obergaulingeret al. 2006; Burrows et al. 2007). As reported by Burrowset al. (2007) these outflows can coexist with equatorial ac-cretion and may not be able to prevent eventual collapse ofPNS. However, they certainly overcome accretion in the po-lar direction and create favorable conditions for activationof the BZ mechanism. The very fast rotation of progenitor’sFe core required in this scenario, with the period around fewseconds, is unlikely to be achieved in stars with strong fossilmagnetic field (see Sec.5.2).

6 In particular, Proga et al. (2003) state that neutrino cooling

is dominated by pair annihilation whereas we find that it is the

lepton capture on free nucleons which is dominant in our models.

A related issue is whether a significant fraction of fos-sil magnetic flux available in the progenitor can actually beaccumulated by the black hole. The traditional viewpointis that the inward advection of magnetic field in accretiondisk is effective and can result in build-up of poloidal mag-netic field up to the equipartition with the disk gas pres-sure (Bisnovatyi-Kogan & Ruzmaikin 1974, 1976; Bland-ford & Znajek 1977; Macdonald & Thorne 1982; Begelmanet al. 1984). On the other hand, it has also been arguedthat in standard viscosity driven accretion disk the turbu-lent outward diffusion of poloidal field may easily neutralisethe inward advection well before it reaches the equipartitionstrength(Lubow et al. 1994; Heyvaerts et al. 1996; Ghosh &Abramowicz 1997; Livio et al. 1999). However, the disk an-gular momentum can also be removed by magnetic stressesvia interaction with the disk corona and wind and this maychange the balance in favour of the inward accretion ofpoloidal magnetic flux (Spruit & Uzdensky 2005; Robstein& Lovelace 2008). This has been demonstrated in recent3D Newtonian simulations of radiatively inefficient accretiondisks where the poloidal magnetic field reached the equipar-tition strength near the central “black hole”(Igumenshchev2008). In our simulations, a significant fraction of blackhole’s magnetic flux is accumulated before the developmentof accretion disk. However, the flux does not escape afterthe disk is formed. On the contrary, it keeps increasing untilthe oppositely directed magnetic field lines begin to enterthe accretion shock. Thus, our results are in agreement withthose by Igumenshchev (2008) and suggest strong magneticinteraction between the disk and its corona/wind.

The alternative origin of magnetic field is via dynamoprocess in the turbulent accretion disk. In this case the sat-urated equilibrium magnetic field strength can be estimatedfrom the energy equipartition argument

B2t

8π' ρv

2t

2, (16)

whereBt and vt are the turbulent magnetic field and velocityrespectively. However, the typical scale of this field is rathersmall, of the order of the disk hight, h rg. When magneticloops of such small scale are advected onto the black hole onewould expect quick, on the time scale of rg/c, reconnectionand annihilation of magnetic field, resulting in low residualmagnetic flux penetrating the black hole. Tout & Pringle(1996) argued that similar reconnection processes in the diskcorona may lead to inverse cascade of magnetic energy onscales exceeding h. The expected spectrum in this range isBλ ∝ 1/λ and thus on the scale of rg one could find theregular magnetic field of strength

Bp 'h

rgBt.

The standard theory of α-disks gives

vt ' αΩkh,

where Ω is the Keplerian frequency, h is the half-thickness ofthe disk and α = 0.1÷0.01 is a free parameter of this theory.According to the theory of thin neutrino cooled accretiondisks by Popham et al. (1999)

h

r= 0.09

(α

0.1

)0.1(Mh

M

)−0.1(r

rg

)0.35

c© 0000 RAS, MNRAS 000, 000–000

Activation of the BZ mechanism 13

and

ρ = 2.4× 1015(α

0.1

)−1.3(Mh

M

)−1.7(r

rg

)−2.55(Ms

M

)g

cm3.

Putting all these equations together we find the typical blackhole magnetic flux

Ψh = 3× 1026(α

0.1

)0.55(Mh

M

)0.95(Ms

M

)0.5

G cm2. (17)

The corresponding power of BZ mechanism is

EBZ ' 1.3× 1050f2(a)(α

0.1

)1.1(Ms

M

)erg s−1 (18)

almost independently on the black hole mass. One can seethat even for maximally rotating black holes one should notexpect much more than 1051erg from a collapse of a 10Mprogenitor. On the other hand, in this model one should ex-pect a much more powerful magnetically driven wind fromthe accretion disk (Livio et al. 1999). Mass loading of thiswind is likely to be much higher due to the processes similarto solar coronal mass injections are the terminal Lorentz fac-tors are not expected to be as high as required from the GRBobservations. However, the disk wind could be responsiblefor the ∼1052ergs required to drive hypernova. Having saidthat, we need to warn that the above equations are purelyNewtonian and thus their application to the very vicinity ofblack hole may introduce large errors. Moreover, the currenttheory of magnetic dynamo in accretion disks can hardly becalled well developed and so one could expect few surprisesas our understanding of the dynamo improves. Finally, theabove picture is in conflict with our numerical models, par-ticularly those which do not show BZ driven explosions. Itis difficult to pin down the exact reasons for this but thekey issues are likely to be the condition of axisymmetry andnumerical resolution.

5.2 The nature of rotation

With a typical rotation rate of 200 km/s on the zero-agemain-sequence (e.g. Penny 1996; Howarth et al. 1997) thestandard evolutionary models of solitary stars predict veryrapid rotation of stellar cores by the time of collapse (Hegeret al. 2000; Hirschi et al. 2004, 2005). According to thesemodels the typical periods of newly born pulsars in suc-cessful supernovae are around one millisecond and “failedsupernovae” conceive rapidly rotating black holes with ac-cretion disks, as required for GRBs. These results, however,contradict to observations which clearly indicate that youngmillisecond pulsars and GRBs are much more rare.

When the magnetic torque due to the relatively weakmagnetic field generated via the Tayler-Spruit dynamo(Tayler 1973; Spruit 2002) is included in the evolutionarymodels they result in significant spin down of stellar coresduring the red giant phase and during the intensive massloss period characteristic for massive stars at the Wolf-Rayetphase (Heger et al. 2005). As the result the predicted rota-tion rates for newly born pulsars come to agreement withthe observations but on the other hand, the core rotationbecomes too low compared to what is required for the col-lapsar model of GRBs (Heger et al. 2005). Recently, it wasproposed that low metalicity could be the factor that allows

to reduce the efficiency of magnetic braking in GRB pro-genitors (Woosley & Heger 2006; Yoon et al. 2006). On onehand, the mass-loss rate of WR-stars decreases significantlywith metalicity. On the other hand, the evolutionary modelsof Woosley & Heger (2006) show that low metalicity stars ro-tating close to break-up speed may become fully convective,chemically homogeneous, and by-pass the red giant phase.This result has been confirmed by Yoon et al. (2008) buttheir models also show that even stars with metalicity twothousand times below that of Sun inflate above 50 solar radiiby the stage of carbon burning. For a potential progenitorof even long duration GRBs the corresponding light-crossingtime is uncomfortably high, tls > 120s. Indeed, this makesvery difficult to explain the large observed fraction of longGRBs with the gamma ray burst duration of only 3-30 sec-onds. It is somewhat curious but the high mass loss rates ofmassive stars with solar metalicity has long been consideredas an important factor in the evolution of GRB progeni-tors as it can result in the formation of a compact heliumstar (WR-star) via total loss of extended hydrogen envelop(Woosley 1993). Obviously, these complications are not spe-cific to the standard fireball model of GRBs. In fact, themagnetic field of magnetic stars is much stronger than thatpredicted by the Taylor-Spruit theory of stellar dynamo andthus the magnetic braking of GRB progenitors is more se-vere in our magnetic model of GRB central engine whichrelies on strong fossil magnetic field.7

To a large extent the above complications are specificto solitary stars and can be avoided in close binaries. Firstly,the extended envelope of a giant star is only weekly boundedby gravity and is easily dispersed in close binary, predomi-nantly in the orbital plane (Taam & Sandquist 2000). Sec-ondly, even in the case of high metalicity and/or strong mag-netic field the stellar rotation rate may remain high due tothe synchronisation of spin and orbital motions. Indeed, theminimum separation between the components of binary star,Lc, is determined by the size of the critical Roche surfacesas

Lc ' 2.64Rq0.2, (19)

where R is the radius of more massive component and qis the ratio of component masses (q ≤ 1). The rotationalperiod is given by

Ω2 = (1 + q)GM

L3, (20)

where L is the actual separation of component and Mis the mass of more massive component. Using q = 1,L = 3R, R = 5R and M = 20M we find that thespecific angular momentum on the stellar equator is about6×1018cm2 s−1. On the other hand the specific angular mo-mentum of marginally bound orbit around Schwarzschildblack hole is

lmb = 4GMh

c, (21)

For Mh = 20M this gives us only 3.5× 1017cm2 s−1. Thus,when the star collapses one would expect the black hole

7 The magnetic field of magnetic stars is most likely to be in-herited from the interstellar medium during star formation (Moss

1987; Braithwaite & Spruit 2004).

c© 0000 RAS, MNRAS 000, 000–000

14 S.S. Komissarov & M.V.Barkov

to develop substantial rotation and to get surrounded byan accretion disk towards the final phase of the collapse(Barkov & Komissarov in preparation). The lower accretionrates expected in this phase make the neutrino mechanismmuch less effective but do not have much of effect on the BZmechanism. Indeed, assuming the equipartition of gas andmagnetic pressure we find that

B2 = 8πp = 8πα−2ρv2t .

Applying this at the inner edge (r = rg) of neutrino-cooledaccretion disk (Popham et al. 1999) we can estimate themaximum magnetic flux that can be kept on the black holeby the disk advection as

Ψh = 3.3× 1028(α

0.1

)−0.55(Mh

M

)1.05(Ms

M

)0.5

G cm2. (22)

For Mh = 20M and M = 0.01M/s this gives us theformidable Ψh ' 7 × 1028G cm2. According to eq.3 muchlower fluxes are needed to explain the observed energetics ofGRBs.

Even faster rotation is expected in the case of mergerof helium star with its compact companion, neutron star ora black hole, during the common envelope phase of a closebinary (Tutukov & Iungelson 1979; Frayer & Woosley 1998;Zhang & Fryer 2001). Since the orbital angular momentumof binary star is

J2orb(L) =

GM2hM

2L

Mh +M. (23)

where Mh is the mass of the compact star, we can roughlyestimate the specific orbital angular momentum depositedat the distance r from the centre of the helium star as

l2(r) = J2orb(r)/M2(r) =

GM2h r

Mh +M(r), (24)

where M(r) is the helium star mass within radius r. WhenM(r) is a slow function, as it is in Bethe’s model whereM(r) ∝ ln r, we may use in eq.24 the total mass of thehelium star, M , in place of M(r). For Mbh = 2M andMhc = 20M this gives us

l(r9) = 1.5× 1017√r9 cm2s−1, (25)

where r9 = r/109cm. In spite of all the approximations, suchestimates agree quite well with the results of numerical sim-ulations (Zhang & Fryer 2001). For r9 > 1 this is higher thanthe upper value of l = 1017 cm2s−1 imposed in the collapsarsimulations by MacFadyen & Woosley (1999). The higherangular momentum is one of the factors that reduces thedisk accretion rates and hence the efficiency of the neutrinoannihilation mechanism (Zhang & Fryer 2001) but, as wehave already commented, for the Blandford-Znajek mecha-nism this is much less of a problem.

The compact companion may initially be a neutron starbut during the in-spiral it is expected to accumulate substan-tial mass and become a black hole (Zhang & Fryer 2001).In fact, during the inspiraling the compact companion ac-cumulates not only mass but spin as well. The final valueof the black hole rotation parameter computed in Zhang &Fryer (2001) varies from model to model between a = 0.14to a = 0.985. Thus, provided that the normal star of thecommon envelope binary is a magnetic star we have all thekey ingredients for a successful magnetically driven GRB

explosion: (i) a rapidly rotating black hole, (ii) a rapidly ro-tating and (iii) compact collapsing star, and (iv) a “fossil”magnetic field which is strong enough to tap the black holeenergy at rates required by observations (see eq.3). It hasbeen noted many times that the magnetic fluxes of magneticAp stars, magnetic white dwarfs, and pulsars are remarkablysimilar (e.g. Ferrario & Wickramasinghe 2005), suggestingthat they could be related. Now one may consider addinganother type of objects to this list - GRB collapsars.

6 CONCLUSIONS

In this paper we continued our study into the potential of theBlandford-Znajek mechanism as main driver in the centralengines of Gamma Ray Bursts. In particularly, we analysedthe conditions for activation of the mechanism in the col-lapsar model where accumulation of magnetic flux by thecentral black hole is likely to be a byproduct of stellar col-lapse.

It appears that the rotating black hole begins to pumpenergy into the surrounding and overpowers accretion onlywhen the rest mass energy density of matter drops belowthe energy density of the electromagnetic field near the er-gosphere (see eq.4). Only under this condition the MHDwaves generated in the ergosphere can transport energy andangular momentum away from the black hole.

In the case of spherical accretion the above criterion canbe written as a condition on the total magnetic flux accu-mulated by the black hole and the total mass accretion rate(see eq.5). However, in the case of nonspherical accretion,characteristic for the collapsar model, this integral condi-tion becomes less applicable due to the spacial separation ofmagnetic field and accretion flow inside the accretion shock.In fact, our numerical simulations show explosions for sig-nificantly lower magnetic fluxes compared to those expectedfrom the integral criterion. We have not included the neu-trino heating in our numerical models. This is another factorthat can make activation of the BZ mechanism in the col-lapsar setup easier.

Both the general energetic considerations and the con-dition for activation of the Blandford-Znajek mechanism re-quire the central black hole to accumulate magnetic fluxcomparable to the highest observed values for magneticstars, Ψ ' 1027G cm2. Current evolutionary models of mas-sive solitary stars indicate that fossil field of such strengthshould slow down the rotation of helium cores below therate required for formation of accretion disks around blackholes of failed supernovae. The conflict between strong fossilmagnetic field and rapid stellar rotation required by the BZmechanism may be resolved in the models where the GRBprogenitor is a component of close binary. In this case thehigh spin of the progenitor can be sustained at the expenseof the orbital rotation. Moreover, the strong gravitationalinteraction between components of close binary helps to ex-plain the compactness of GRB progenitors deduced from theobserved durations of bursts.

The magnetic field can also be generated in the accre-tion disk and its corona but in this case the magnetic fluxthrough the black hole ergosphere is not sufficiently highto explain the power of hypernovae by the BZ mechanismalone. On the other hand, in this case the disk is expected to

c© 0000 RAS, MNRAS 000, 000–000

Activation of the BZ mechanism 15

drive MHD wind more powerful than the BZ jet. Althoughsuch a wind is unlikely to reach the high Lorentz factorsdeduced from the observations of GRB jets (due to highmass loading) it could be behind the observed energetics ofhypernovae.

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APPENDIX A: DUST ACCRETION INKERR-SCHILD COORDINATES

The problem of dust accretion has been considered in Misneret al. (1973). In the coordinate basis of the Boyer-Lindquist

coordinates, ∂/∂xν′ the components of 4-velocity of a dust

particle accreting from rest at infinity and having zero an-gular momentum are

ut′

= 1 + ζ(r2 + a2)/∆;

uφ′

= 2aζ/∆;

ur′

= −ζη;

uθ′

= 0,

(A1)

where η =√

(r2 + a2)/2r. In order to find the 4-velocitycomponents in the coordinate basis of Kerr-Schild coordi-nate one can simply use the corresponding transformationlaw (Komissarov 2004a):

dt = dt′ + (2r/∆)dr′;dφ = dφ′ + (a/∆)dr′;dr = dr′;dθ = dθ′.

(A2)

This gives us

ut = 1 + ζη/(1 + η);

uφ = −a/(A(1 + η));ur = −ζη;

uθ = 0.

(A3)

Notice that in both coordinate systems the particles moveover a conical surface, θ =const. Their rotation about thesymmetry axis is due to the inertial frame dragging effect.

In order to find the density distribution consider thesteady-state version of the continuity equation:

∂r(√−gρur) = 0,

where g = − sin2 θA2 is the determinant of the metric tensorin the Kerr-Schild (and Boyer-Lindquist) coordinates. Thisshows that

√−gρur = C(θ),

where C(θ) is some arbitrary function and then that

ρ(r, θ) =C(θ)

sin θ√

2r(r2 + a2).

At infinity this function does not depend on the polar angleonly if C ∝ sin θ which implies that ρ does not depend on θfor any r. Denoting as ρ+ the rest mass density at the outerevent horizon we may now write

ρ = ρ+

(r+r

)1

η. (A4)

APPENDIX B: INITIAL MAGNETIC FIELD

In this section we employ the 3+1 splitting of electrodynam-ics described in Komissarov (2004a) where the electromag-netic field is represented by four 3-vectors, D,E,B, and H,defined via

Bi = α ∗F it, (B1)

Ei =α

2eijk

∗F jk, (B2)

Di = αF ti, (B3)

Hi =α

2eijkF

jk, (B4)

where ∗Fµν is the Faraday tensor, Fµν is the Maxwell ten-sor, eijk is the Levi-Civita tensor of space and α is the lapsefunction. In stationary metric, ∂tgνµ = 0, the evolution ofthese vector fields is described by the Maxwell equations forelectromagnetic field in matter. The effects of curved spacetime are incorporated via the non-Euclidian spatial metricand the constitutive equations

E = αD + β×B, (B5)

H = αB − β×D. (B6)

where α is the lapse function and β is the shift vector. Interms of these vectors the condition of perfect conductivity,Fνµu

µ = 0, can be written in the familiar form

E = −v×B, (B7)

where vi = dxi/dt is the usual 3-velocity vector.Both in the Boyer-Lindquist and the Kerr-Schild coordi-

nates the poloidal component of Poynting flux, Si = −αT it ,is given by

Sp = Ep×Hφ +Eφ×Hp (B8)

whereas the the poloidal component of the angular momen-tum flux, Li = αT iφ, is

Lp = −(E ·m)Dp − (H ·m)Bp, (B9)

where m = ∂/∂φ.Suppose that in the Boyer-Lindquist coordinates the

magnetic field is purely radial, B′θ = B′φ = 0. Since β ‖meq.B6 then implies that

H ′φ = αB′φ = 0.

Provided this magnetic field is frozen into the flow describedby eq.A1 we also find that

E′ = −v′φ×B′

r.

Given these results the equations B9 and B8 ensure that

S′p = L′p = 0.

In order to find the corresponding electromagnetic fieldin the Kerr-Schild coordinates we simply apply the transfor-mation law A2). This way we find that

Sp = Lp = 0 (B10)

and

Bθ = 0, Bφ = Bra(A+ 2r)

A∆ + 2r(r2 + a2). (B11)

c© 0000 RAS, MNRAS 000, 000–000

Activation of the BZ mechanism 17

The divergence-free condition requires

∂r(√γBr) = 0

and hence

Br =f(θ)√γ

where γ is the determinant of the spatial metric. Providedthat at infinity (where γ ∝ sin2 θ) Br does not depend onthe polar angle we then have

Br = B0sin θ√γ. (B12)

c© 0000 RAS, MNRAS 000, 000–000